Advanced Geometry 2017–2018 ENS de Lyon
Frobenius Theorem
Exercise 1. LetX: (x, y, z)7→ ∂x∂ −y∂z∂ andY : (x, y, z)7→ ∂y∂ be two vectors fields on R3. 1. Prove that∀p∈R3,(X(p), Y(p))is linearly independent.
2. Compute[X, Y]. Is the distribution spanned by X andY integrable?
3. Recover this result without using Frobenius Theorem.
Exercise 2. LetX: (x, y)7→x∂x∂ +y∂y∂ etY : (x, y)7→x∂y∂ −y∂x∂ be two vector fields onR2. 1. Compute[X, Y]. Are there coordinates(s, t) on some neighborhood of(1,0)such that
X = ∂s∂ and Y = ∂t∂?
2. Compute the flows ofX and Y.
3. Build explicit coordinates (s, t) on some neighborhood of (1,0)such that X = ∂s∂ and Y = ∂t∂.
1
